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➗ Math · Numbers and Algebra · Lesson 5 of 8 · 6 min

Solving one-variable linear equations

⚖️ An equation is a balance scale. Do the same thing to both sides and you can always get the variable alone.

💡 Key idea

Undo operations in reverse order, do the same thing to both sides, then substitute to check.

🧠 The idea

To SOLVE an equation means to find the value of the variable that makes both sides equal. Use INVERSE OPERATIONS to undo what was done to the variable, in reverse order: undo addition or subtraction first, then multiplication or division. Whatever you do to one side, do to the other. For 3x + 7 = 25, subtract 7 from both sides (3x = 18), then divide both sides by 3 (x = 6). If the variable appears on both sides, collect the variable terms on one side first: 5x − 4 = 2x + 11 becomes 3x = 15, so x = 5. If there are parentheses, DISTRIBUTE: 2(x + 3) = 2x + 6. With fractions, multiply both sides by the denominator to clear them: x/4 = 9 gives x = 36. Always CHECK by substituting your answer into the original equation.

✏️ Worked example

Solve 4(x − 2) + 3 = 2x + 9.
Step 1: Distribute: 4x − 8 + 3 = 2x + 9, so 4x − 5 = 2x + 9.
Step 2: Subtract 2x from both sides: 2x − 5 = 9.
Step 3: Add 5 to both sides: 2x = 14.
Step 4: Divide both sides by 2: x = 7.
Step 5: Check: 4(7 − 2) + 3 = 4 × 5 + 3 = 23, and 2 × 7 + 9 = 23. ✓
Answer: x = 7.

📌 Takeaways

  • Undo + and − first, then × and ÷.
  • Distribute parentheses and gather variable terms on one side.
  • Substitute your answer back in to check it.

📖 Terms in this lesson

Variable: A letter that stands for an unknown or changing number; for example, in 2x + 5 = 11, x = 3.

✅ Check yourself

Question 1 of 3

Solve 5x − 8 = 27.

  1. x = 19
  2. x = 175
  3. x = 3.8
  4. x = 7
Show the answer and rationale

Answer: D

Add 8 to both sides: 5x = 35. Divide by 5: x = 7. The answer 3.8 subtracts 8 instead of adding it (19 ÷ 5), 19 makes that same slip and also forgets to divide by 5, and 175 multiplies by 5 instead of dividing.

Question 2 of 3

Solve 3(x + 4) = 33.

  1. x = 7
  2. x = 9.67
  3. x = 29
  4. x = 11
Show the answer and rationale

Answer: A

Divide both sides by 3: x + 4 = 11, so x = 7. Check: 3(7 + 4) = 33. The answer 9.67 multiplies only the x by 3 (3x + 4 = 33), and 11 stops before subtracting the 4.

Question 3 of 3

Solve 7x + 2 = 4x + 20.

  1. x = 18
  2. x = 2
  3. x = 6
  4. x = 7.33
Show the answer and rationale

Answer: C

Subtract 4x from both sides: 3x + 2 = 20. Subtract 2: 3x = 18. Divide by 3: x = 6. The answer 2 comes from adding the x terms and the constants (11x = 22) instead of moving them to opposite sides, 18 forgets to divide by 3, and 7.33 adds the 2 instead of subtracting it (3x = 22).

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